Pretty much the title.
Well, a Beast Mastery Hunter is a soloing machine, with the ability to tank, DPS, and heal a bit too. they can continuously engage and defeat packs of mobs including elites.
But if you are looking for a partner, I would look for one that can either do damage but let your pet keep agro (like a Demonology Warlock), or one that can buff your pet and yourself (like a Shaman with totems or a Paladin with blessings), or a class that can hold agro so you can go for a DPS pet or perhaps soec into Marksman. The overall idea being, your pet is a primary feature of your class, so pick a class that works well with it, one way or another.
I would not recommend just one class in particular, because the person leveling with you should decide that. They may want to try class X, or hate class Y, or want the ability to tank or heal when they hit 80, etc.
As long as there is good synergy between the roles fulfilled, any class can level with a hunter. Think of yourself as either the "tank" or "DPS" role when leveling. (I wouldn't recommend actually trying to tank an instance as a hunter.) If you want to hold all the aggro, get a Tenacity pet and spec mostly into Beast Mastery. This will allow a healer/DPS partner to focus on keeping you and your pet up or killing without fear of pulling aggro. If you want to be dealing the most DPS and don't like Beast Mastery or Tenacity pets, partner with a class that is capable of taunting and holding aggro so that you can unload on the target.
Ultimately, it is up to you and your partner. I leveled pre-BC 1-58 through TBC release 58-70 as a full Marksmanship hunter with a crab as a pet. My leveling partner was a Balance druid. Neither of those are optimal leveling builds for those classes, but we really didn't have any trouble - and this was before all of the buffs and changes to Balance druids and pet talent trees and whatnot. You should have no trouble as long as you keep basic role synergy in mind.
Feral Druid. Can throw your pet a hot or two when needed, then shift back into cat to continue dps'ing, or bear for tanking the tougher stuff while hunter/pet dps. Two powerful buffs, plus Leader of the Pack and Improved Leader of the Pack which afaik buff both hunter and pet.
Other healing hybrids like pallies and shammans are good too, but druids are the most versatile w/ the least downtime, and their array of hots give them the ability to heal your pet while continuing to dps. Druids also have the least downtime of any class except perhaps Blood DK's, so it won't slow your hunter down.
from my own experience levelling a hunter pretty much any class will work with you being that you have your own tank in your pet. I think the best would be a paladin for the ability to throw a heal, buff and also tank should your pet go down. Yes a feral druid can also do that but as you level you will have conflicts over gear drops, even after level forty you'll still find attractive leathes options you might want. With a paladin you wont have these issues. Same applies to an enhancement shaman, you'll have gear conflicts all the way to eighty. At the end of the day hunters are the perfect solo class available so you dont need a partner at all.
I leveled to 80 with me playing a prot Pally and my wife playing a Hunter. It's a good combo and we were even able to duo some dungeons while the quests were still green. AE is fun with that pair as well. I still think a Tank/Healer makes the best duo (my wife and I also play a warrior shaman pair who just cannot die), but a self healing tank like a Pally, Druid or even dk can work well with a hunter and you can take on some tough challenges with that pair. The main downside is when you attempt challenging things like duping dungeons the tank has to self heal.
I used Refer-A-Friend to level a Feral Druid up with my friends BM Hunter. The Hunter had a Tenacity pet that held agro and I went Feral DPS. We flew through the levels (partly because of the RAF bonus), but we also had almost zero down time. The BM Hunter could heal his pet easily and I could keep both of us healed up. If we got over our heads and the hunter died I could rez him which was a big time saver. With the +5% Crit from Leader of the Pack, we were pulling very solid DPS numbers.
Assuming you are looking at speed leveling...
I'd definitely recommend a DPS Hybrid with a heal button. If you can both DPS targets with the second toon casting the odd heal you'll find that you can rip through most non-dungeon content. Dual specs for the hybrid (switching to heal or even tank) help with nasty situations or +level quests / mobs.